I asked this question [here][1], but received no answer.

Recently, reading [this problem][2], I found out that

$ \lim_{n\to \infty} \int_{0}^{1} \cdots \int_{0}^{1} \frac{x_1^q + \cdots + x_n^q}{x_1^p + \cdots + x_n^p} \, \mathrm{d}x_1 \cdots \mathrm{d}x_n
=\frac{p+1}{q+1} $

Is it known a general formula for that multiple integral when we fix $ n, p $ and $q$ as positive integers? Otherwise is it possible to have a general formula if $q=2, p=1$ and $n$ is a fixed positive integer?


  [2]: https://math.stackexchange.com/questions/4801426/lim-n-to-infty-int-01-cdots-int-01-fracx-1qx-2q-cdotsx-nqx-1px


  [1]: https://math.stackexchange.com/questions/4906414/general-formula-for-int-01-cdots-int-01-fracx-1q-cdots-x